Question 2: The textbook provides a summary of how to find the limits of integration when integrating in the order dy dx. Then, the textbook says for the reverse order to “use horizontal lines instead of vertical lines in Steps 2 and 3 (see Figure 15.15).” Go through this process: explain each step of the procedure for finding the limits of integration when integrating in the order dx dy.Hint 1: Read the subsection "Finding Limits of Integration" (p. 906). RLSmallest xis x = 0(b)Leaves aty=√1-²Enters aly=1-xLargest xis x = 1(c)FIGURE 15.14 Finding the limits ofintegration when integrating first withrespect to y and then with respect to .x.Using Vertical Cross-Sections When faced with evaluatingf(x, y) dA, integrat-ing first with respect to y and then with respect to x, do the following three steps:1. Sketch. Sketch the region of integration and label the bounding curves (Figure 15.14a).2. Find the y-limits of integration. Imagine a vertical line L cutting through R in the direc-tion of increasing y. Mark the y-values where L enters and leaves. These are the y-limitsof integration and are usually functions of x (instead of constants) (Figure 15.14b).3. Find the x-limits of integration. Choose x-limits that include all the vertical linesthrough R. The integral shown here (see Figure 15.14c) is[[₁pind provi-g-pp,x=0f(x, y) da=f(x, y) dy dx.Using Horizontal Cross-Sections To evaluate the same double integral as an iteratedintegral with the order of integration reversed, use horizontal lines instead of vertical linesin Steps 2 and 3 (see Figure 15.15). The integral is[[ [²²f(x,y) dA=f(x, y) dx dy. 9060Chapter 15 Multiple IntegralsRRFIGURE 15.13 The region of integrationin Example 2.L_x² + y² = 1R<-1(a)1X1Leaves aty-VI-aEnters aty=1-xxAlthough Fubini's Theorem assures us that a double integral may be calculated as aniterated integral in either order of integration, the value of one integral may be easier tofind than the value of the other. The next example shows how this can happen.EXAMPLE 2 Calculate[[Rwhere R is the triangle in the xy-plane bounded by the x-axis, the line y = x, and the linex = 1.sin.xdA,Solution The region of integration is shown in Figure 15.13. If we integrate first withrespect to y and next with respect to x, then because x is held fixed in the first integration,we findL'inx dy) dx = f[x0- f'sit0If we reverse the order of integration and attempt to calculateIS, dx dy.sin xXsinx²X-X3-0dx =sin.x dx = -cos (1) + 1 ≈ 0.46.we run into a problem because ((sinx)/x) dx cannot be expressed in terms of elemen-tary functions (there is no simple antiderivative).There is no general rule for predicting which order of integration will be the good onein circumstances like these. If the order you first choose doesn't work, try the other. Some-times neither order will work, and then we may need to use numerical approximations.Finding Limits of IntegrationWe now give a procedure for finding limits of integration that applies for many regions inthe plane. Regions that are more complicated, and for which this procedure fails, can oftenbe split up into pieces on which the procedure works.

Using Horizontal Cross-Sections: To evaluate the double integral integrating first with respect to…

R L 0 Smallest x is x = 0 (b) Leaves at y=√1-rª² Enters al y=1-x Largest x is x = 1 (c) FIGURE 15.14 Finding the limits of integration when integrating first with respect to y and then with respect to .x. Using Vertical Cross-Sections When faced with evaluating f(x, y) dA, integrat- ing first with respect to y and then with respect to x, do the following three steps: 1. Sketch. Sketch the region of integration and label the bounding curves (Figure 15.14a). 2. Find the y-limits of integration. Imagine a vertical line L cutting through R in the direc- tion of increasing y. Mark the y-values where L enters and leaves. These are the y-limits of integration and are usually functions of x (instead of constants) (Figure 15.14b). 3. Find the x-limits of integration. Choose x-limits that include all the vertical lines through R. The integral shown here (see Figure 15.14c) is px-1-VI-2² [[ f(x, y) dA = x=0 y=1-x f(x, y) dy dx. Using Horizontal Cross-Sections To evaluate the same double integral as an iterated integral with the order of integration reversed, use horizontal lines instead of vertical lines in Steps 2 and 3 (see Figure 15.15). The integral is If f(x, y) dA= ^= √√√²-²₁ f(x, y) dx dy.

R L 0 Smallest x is x = 0 (b) Leaves at y=√1-rª² Enters al y=1-x Largest x is x = 1 (c) FIGURE 15.14 Finding the limits of integration when integrating first with respect to y and then with respect to .x. Using Vertical Cross-Sections When faced with evaluating f(x, y) dA, integrat- ing first with respect to y and then with respect to x, do the following three steps: 1. Sketch. Sketch the region of integration and label the bounding curves (Figure 15.14a). 2. Find the y-limits of integration. Imagine a vertical line L cutting through R in the direc- tion of increasing y. Mark the y-values where L enters and leaves. These are the y-limits of integration and are usually functions of x (instead of constants) (Figure 15.14b). 3. Find the x-limits of integration. Choose x-limits that include all the vertical lines through R. The integral shown here (see Figure 15.14c) is px-1-VI-2² [[ f(x, y) dA = x=0 y=1-x f(x, y) dy dx. Using Horizontal Cross-Sections To evaluate the same double integral as an iterated integral with the order of integration reversed, use horizontal lines instead of vertical lines in Steps 2 and 3 (see Figure 15.15). The integral is If f(x, y) dA= ^= √√√²-²₁ f(x, y) dx dy.

Calculus For The Life Sciences

2nd Edition

ISBN:9780321964038

Author:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Publisher:GREENWELL, Raymond N., RITCHEY, Nathan P., Lial, Margaret L.

Chapter7: Integration

Section7.EA: Extended Application Estimating Depletion Dates For Minerals

Problem 6EA

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